Find the extreme values (absolute and local) of the function over its natural domain, and where

step 1 We are trying to find the absolute extrema for this function. First, let's use some reasoning to

Find the extreme values (absolute and local) of the function...

Key Concepts

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When analyzing extreme values, it is crucial to understand the natural or inherent domain because it determines the interval over which the optimization is carried out and ensures that any candidates for extreme values are valid within the context of the function.

Differentiation and Derivatives

Differentiation is a central tool in calculus used to determine the rate at which a function changes with respect to its variable. The derivative represents the slope of the tangent line at any point on the function, which helps identify where the function is increasing or decreasing, and thus where potential extreme values may occur.

Critical Points

Critical points are specific points in the domain of a function where the derivative is either zero or undefined. These points are important in the study of optimization because they are the most likely candidates for local minima or maxima, as they often indicate a change in the trend of the function’s graph.

Local and Absolute Extrema

Local extrema refer to the highest or lowest values of a function within a small, neighboring interval, whereas absolute (or global) extrema are the highest or lowest values over the entire domain of the function. Identifying these extrema involves comparing values at critical points and possibly at the endpoints of the domain.

Optimization Techniques

Optimization involves using methods such as the first derivative test and the second derivative test to analyze the behavior of a function around its critical points. These techniques help determine whether a given critical point is a local maximum, a local minimum, or neither, and are key to systematically finding both local and absolute extreme values.

Transcript

00:01 We are trying to find the absolute extrema for this function.

00:05 First, let's use some reasoning to kind of determine what's happening with this graph.

00:11 We do know at the x value of zero, that y is zero.

00:17 I also know that if x is greater than zero, that y is going to be positive.

00:24 Now, how do i know that? well, the denominator is always going to be positive regardless of the x value.

00:30 And if x is greater than zero, that would make the top of the fraction positive as well.

00:35 So it's going up.

00:38 It may come back down again, but i know it's got to remain above the x -axis.

00:44 If x is negative, i know that y is negative.

00:48 And that's because the denominator is going to be positive, since x squared plus one is always positive, but the denominator is going to be a negative.

00:56 So it's going to go below the x -axis.

00:58 I don't know what's happening, beyond that, other than the fact that it's never going to jump over the x -axis.

01:06 So let's find our critical numbers.

01:08 The derivative will come from the quotient rule, bottom, times the derivative of the top, which is one, minus the top, times the derivative of the bottom, 2x, over the denominator squared.

01:25 We can simplify that.

01:27 That's going to be x squared plus 1 minus 2x squared.

01:36 And in the, oops, it's supposed to be squared down here.

01:41 And in the denominator, x squared plus one quantity squared, which simplifies to negative x squared plus one over x squared plus one quantity squared...

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